My objective is called "domain and range,determine from graphs", so by the name of the objective you can probably guess that i will show you how to know what a graph will look like by the way the equation is written.
The first thing to know is what y=x^2 looks like on a graph.
As you can see here on the graph of y=x^2 that the vertex is on the origin (0,0), it opens upwards, and it is standard width.
These kinds of questions only deal with parabolas, y=a(x-h)^2+k. H and K determine where the vertex is, so on y=x^2 notice that there is no h or k so they are both 0. That is why the vertex is on (0,0).
To determine where the vertex of the graph is located you must look at the h and k values. The h value moves it on the x-axis (right and left), if the value is + your move it to the left and if it is - you move it to the right.
Ex. The equation for A is y=(x+2)^2 and the equation for B is y=(x-2)^2. Notice that the only difference is what way the graph moved along the x-axis.
Now that we know what the h value does to the graph we will figure what the k value does to the graph.
The equation for A is y=x^2-1 and the equation for B is y=x^2+1, the only difference in the graph compared to the standard graph of y=x^2 is that it moved up and down on the y-axis.
So to kind of recap what we learnt so far, here is an example.
The equation for this graph is y=(x+3)^2-3 so the vertex of the graph is (-3,-3). Always remember that a positive h value moves the vertex to the left and a nagative value moves it to the right, and if the k value is positive then the vertex moves up and if it is negative it moves down.
Now that we know what the h and k values do we will move onto the a value. First off the sign of the a value, if it is a negative value then the parabola will open downwards and if it is positive it will open upwards, pretty simple. The most difficult and confusing thing is the a value its self. The a value sets the width of the graph, standard width is when there is no A value or 1. If you have a graph with an A value you multiply the Y values on points on the graph by the A value.
So lets say that the points (1,1) and (2,2) are on your graph and you have an a value is 2, so now the points on your new graph would be (1,2) and (2,4), the y value is doubled, if the a value was 3 then the y value is tripled and so on. So basically if the a value is greater then 1 it will get narrower, and if it is any number between 0 and 1 then it will get wider.
ex.
Ok well there are 4 lines on this one graph. Line B is y=x^2 it is standard width. Line C is obviously narrower and the equation for this line is y=2x^2, so on the standard graph where the point is (1,1) the point on graph C is (1,2), the y value is multiplied by the A value . Line A has an equation of y=0.2x^2 so the point is (1,0.2), once again the y value is multiplied by the A value. Now line D is identical to line C except it opens downward, so the difference is equations is one is y=-2x^2 and one is y=2x^2, the - sign in front of the two makes it open downward.
Now that we know what all the values mean you can put it into one equation.
ex.
So now lets say you are working on accelerated math and it gives you a question where it gives you an equation and gives you 4 different graphs to choose from. start off by checking the vertex. Just look at the h and k values, lets say the equation is y=(x-3)^2-2, the vertex has to be (3,-2), so look at the graphs and check it out, you could easily narrow it down, also check the sign in front of the A value, if its positive it opens up and if its negative it opens down. NOw if you still have more then one answer look at the A value, if it is greater then 1 it will be narrow and if it is between 0 and 1 it will be wider.
ex.What is the graph for y=2(x-1)^2+2
A is the correct answer because if you look at the h and k points you can tell where the vertex is, so -1 and 2 means the vertex is at (1,2) so right off the bat it eliminates C and D, now if you look at the value of the A value you will notice that it is positive so the graph has to open upwards so that eliminates B so you have your answer.
If you read this whole wiki you will now beable to tell what the graph of any equation of a parabola will look like.
My objective is called "domain and range,determine from graphs", so by the name of the objective you can probably guess that i will show you how to know what a graph will look like by the way the equation is written.
The first thing to know is what y=x^2 looks like on a graph.
As you can see here on the graph of y=x^2 that the vertex is on the origin (0,0), it opens upwards, and it is standard width.
These kinds of questions only deal with parabolas, y=a(x-h)^2+k. H and K determine where the vertex is, so on y=x^2 notice that there is no h or k so they are both 0. That is why the vertex is on (0,0).
To determine where the vertex of the graph is located you must look at the h and k values. The h value moves it on the x-axis (right and left), if the value is + your move it to the left and if it is - you move it to the right.
Ex.
Now that we know what the h value does to the graph we will figure what the k value does to the graph.
The equation for A is y=x^2-1 and the equation for B is y=x^2+1, the only difference in the graph compared to the standard graph of y=x^2 is that it moved up and down on the y-axis.
So to kind of recap what we learnt so far, here is an example.
The equation for this graph is y=(x+3)^2-3 so the vertex of the graph is (-3,-3). Always remember that a positive h value moves the vertex to the left and a nagative value moves it to the right, and if the k value is positive then the vertex moves up and if it is negative it moves down.
Now that we know what the h and k values do we will move onto the a value. First off the sign of the a value, if it is a negative value then the parabola will open downwards and if it is positive it will open upwards, pretty simple. The most difficult and confusing thing is the a value its self. The a value sets the width of the graph, standard width is when there is no A value or 1. If you have a graph with an A value you multiply the Y values on points on the graph by the A value.
So lets say that the points (1,1) and (2,2) are on your graph and you have an a value is 2, so now the points on your new graph would be (1,2) and (2,4), the y value is doubled, if the a value was 3 then the y value is tripled and so on. So basically if the a value is greater then 1 it will get narrower, and if it is any number between 0 and 1 then it will get wider.
ex.
Ok well there are 4 lines on this one graph. Line B is y=x^2 it is standard width. Line C is obviously narrower and the equation for this line is y=2x^2, so on the standard graph where the point is (1,1) the point on graph C is (1,2), the y value is multiplied by the A value . Line A has an equation of y=0.2x^2 so the point is (1,0.2), once again the y value is multiplied by the A value. Now line D is identical to line C except it opens downward, so the difference is equations is one is y=-2x^2 and one is y=2x^2, the - sign in front of the two makes it open downward.
Now that we know what all the values mean you can put it into one equation.
ex.
So now lets say you are working on accelerated math and it gives you a question where it gives you an equation and gives you 4 different graphs to choose from. start off by checking the vertex. Just look at the h and k values, lets say the equation is y=(x-3)^2-2, the vertex has to be (3,-2), so look at the graphs and check it out, you could easily narrow it down, also check the sign in front of the A value, if its positive it opens up and if its negative it opens down. NOw if you still have more then one answer look at the A value, if it is greater then 1 it will be narrow and if it is between 0 and 1 it will be wider.
ex.What is the graph for y=2(x-1)^2+2
A is the correct answer because if you look at the h and k points you can tell where the vertex is, so -1 and 2 means the vertex is at (1,2) so right off the bat it eliminates C and D, now if you look at the value of the A value you will notice that it is positive so the graph has to open upwards so that eliminates B so you have your answer.
If you read this whole wiki you will now beable to tell what the graph of any equation of a parabola will look like.